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6 – Greek Math After Euclid. The student will learn about. Greek mathematics after the time of Euclid. §6-1 Historical Setting. Student Discussion. §6-2 Archimedes. Student Discussion. §6-2 Archimedes 1. 1. Classical method of determining . 2. Quadrature of a parabola – more follows. - PowerPoint PPT Presentation
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6 – Greek Math After Euclid
The student will learn about
Greek mathematics after the time of Euclid.
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§6-1 Historical Setting
Student Discussion.
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§6-2 Archimedes
Student Discussion.
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§6-2 Archimedes 1
1. Classical method of determining .2. Quadrature of a parabola – more follows.3. Spiral of Archimedes, r = k .
4. Spheres and cylinders – more follows.
5. Conchoids and Spheroids6. Sand reckoning.7. Plane Equilibrium – centroids.
8. Floating bodies – hydrostatics.
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§6-2 Archimedes 2Quadrature of a parabola – area of a parabolic segment is four-thirds that of the inscribed triangle having the same base.
1. Use calculus to calculate the area bounded by the parabola y = x2 and y = 4 for –2 x 2.
2. Use Archimedes’ method to calculate the area bounded by the parabola y = x2 and y = 4 for –2 x 2.
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§6-2 Archimedes 3
Spheres and cylinders
1. Confirm that the surface area of a sphere is equal to 2/3 the surface area of a circumscribed cylinder.
2. Confirm that the volume of a sphere is equal to 2/3 the volume of a circumscribed cylinder.
4 r 2 = 2/3 (6 r 2)
4/3 r 3 = 2/3 (2 r 2)
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§6-3 Eratosthenes
Student Discussion.
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§6-4 Apollonius
Student Discussion.
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§6-4 Apollonius 1
Problem of Apollonius – Given three circles (degenerate cases permitted) construct a circle tangent to the given circles.
Given three points - easy
Other cases vary in difficulty.Given three lines - easy
If time do two points and a line.
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§6-4 Apollonius 2
From “Plane Loci”.
If A and B are fixed points and k is a given constant, then the locus of a point P, such that AP/BP = k is either a circle (if k 1) or a straight line (if k = 1.).
1. Case where k = 1.
2. Case where k = 2.
The locus is the perpendicular bisector of AB.
Homework: Describe the locus circle completely.
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§6-5 Hipparchus, Menelaus, Ptolemy, and Trigonometry.
Student Discussion.
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§6-5 Hipparchus
The chord of 36 = 37; 04, 55
Chord 36 = 2 · 60 · sin 18
sin 18 = chord 36 / 120
Too small by 0.0000008 or the thickness of a human hair over the length of a soccer field.
sin 18 = 37; 04, 55 / 120
sin 18 = 18; 32, 27, 30 / 60sin 18 = 00 ; 18, 32, 27, 30
sin 18 = 0.309016204ten
18 18
x60
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§6-5 Menelaus.
Menelaus’ Theorem.If transversal LMN intersects the three sides of a triangle then:
1NA
CN
MC
BM
LB
AL
A B
C
L
M
N
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§6-5 Ptolemy.
Three Point Problem.
Given points A, B, and C, and angles AVB, AVC, and BVC, find point V.
. A
. B
. C
. V ?
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§6-6 Heron
Student Discussion.
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§6-6 Heron
1. Area of a triangle of sides a, b, and c, is:
)cs)(bs)(as(s
2. Square root approximation –
If a1 is an approximation of the square root of n then
2
an
a
a 11
2
is a better approximation.
Try 45.
Note: the Babylonians used this for 2.
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§6-7 Ancient Greek Algebra
Student Discussion.
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§6–8 Diophantus
Student Discussion.
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§6–9 Pappus
Student Discussion.
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§6–10 The Commentators
Student Discussion.
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Time Line
2400-1600-525 B.C. Babylonians1900-1000-000 B.C. Egyptions600 B.C. Thales
540 B.C. Pythagoras
450 B.C. Zeno440 B.C. 2 Irrational390 B.C. Socrates / Plato
336-323 B.C. Alexander the Great’s Reign
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Time Line
300 B.C. Euclid287-212 B.C. Archimedes230 B.C. Eratosthenes
225 B.C. Apollonius
44 B.C. Death of Julius Caesar
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Time Line
150 Ptolemy250 Diophantus300 Pappus
390 Theon of Alexandria
410 Hypatia529 School of Athens closed
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Assignment
Read Chapter 7.
